## Provability logic (2004)

Venue: | Handbook of Philosophical Logic, 2nd ed |

Citations: | 27 - 9 self |

### BibTeX

@INPROCEEDINGS{Artemov04provabilitylogic,

author = {Sergei N. Artemov and Lev D. Beklemishev},

title = {Provability logic},

booktitle = {Handbook of Philosophical Logic, 2nd ed},

year = {2004},

pages = {229--403},

publisher = {Kluwer}

}

### Years of Citing Articles

### OpenURL

### Abstract

The idea of provability logic seems to originate in a short paper [Gödel, 1933]. K. Gödel was motivated by the question of providing Brouwer’s intuitionistic logic, as formalized by Heyting, with an adequate semantics. According to Brouwer, intuitionistic truth means provability. Here is a

### Citations

1567 |
Reasoning about Knowledge
- Fagin, JY, et al.
- 1995
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Citation Context ...s is adopted: F is known ∼ F holds in all possible situations. (1) The resulting Epistemic Logic has been remarkably successful in terms of developing a rich mathematical theory and applications (cf. =-=[20; 44]-=-, and other sources). However, the notion of justification, which has been an essential component of epistemic studies, was notoriously absent in the mathematical models of knowledge within the episte... |

755 |
Introduction to Metamathematics
- Kleene
- 1964
(Show Context)
Citation Context ...Axioms and rules of LP reflect the properties of the proof predicates in formal mathematical theories. I. Axioms of classical propositional logic Standard axioms of classical logic, e.g., A1-A10 from =-=[37]-=- II. Axioms of the Logic of Proofs LP s:(F →G) → (t:F →(s·t):G) (Application) t:F → !t:(t:F ) (Proof Checker) s:F →(s+t):F , t:F →(s+t):F (Monotonicity) t:F →F (Reflection) III. Rules of inference F, ... |

232 | Many-valued modal logics
- Fitting
- 1992
(Show Context)
Citation Context ...ket’ has been confirmed in our formal model of Case I. 6.2 Eliminating Definite Description, Russell’s style We can eliminate definite descriptions from Case I using, e.g., Russell’s translation (cf. =-=[24; 48; 54; 55]-=-) of definite descriptions. According to Russell, C(ιxJ(x)) contains a hidden uniqueness assumption and reads as ∃x[J(x) ∧ ∀y(J(y)→y = x) ∧ C(x)], (20) and Jones = ιxJ(x) as J(Jones) ∧ ∀y(J(y)→y = Jon... |

225 |
Philosophical Explanations
- Nozick
- 1981
(Show Context)
Citation Context ...s Justified True Belief in the dialogues Theaetetus and Meno was widely accepted until 1963 when a paper by Edmund Gettier [26] opened the door to a broad philosophical discussion of the subject (cf. =-=[17; 29; 42; 49; 57]-=- and many others). Meanwhile, commencing from seminal works [34; 62], the notions of Knowledge and Belief have acquired formalization by means of modal logic with atoms KF (F is known) and BF (F is be... |

212 |
On denoting
- Russell
- 1905
(Show Context)
Citation Context ...ket’ has been confirmed in our formal model of Case I. 6.2 Eliminating Definite Description, Russell’s style We can eliminate definite descriptions from Case I using, e.g., Russell’s translation (cf. =-=[24; 48; 54; 55]-=-) of definite descriptions. According to Russell, C(ιxJ(x)) contains a hidden uniqueness assumption and reads as ∃x[J(x) ∧ ∀y(J(y)→y = x) ∧ C(x)], (20) and Jones = ιxJ(x) as J(Jones) ∧ ∀y(J(y)→y = Jon... |

164 | Basic Proof Theory
- Troelstra, Schwichtenberg
- 2000
(Show Context)
Citation Context ...ory of justification has roots in the mathematical theory of proofs. It can be traced back to Brouwer-Heyting-Kolmogorov informal semantics for intuitionistic logic [60], typed combinatory logic (cf. =-=[59]-=-), Kleene realizability semantics for intuitionistic logic [36], Gödel’s logic of provability [27], and Logic of Proofs [2; 4; 28]. There are several natural interpretations of Justification Logic. Ju... |

129 | Belief, awareness and limited reasoning
- FAGIN, HALPERN
- 1988
(Show Context)
Citation Context ... in formal epistemology. However, the principle (12) smuggles the logical omniscience defect into modal epistemic logic, which does not have the capacity to measure how hard it is to attain knowledge =-=[18; 19; 35; 47; 51]-=-. Justification Logic provides natural means of escaping logical omniscience by keeping track of the size of evidence terms [10]. 3.2 Monotonicity of Justification The Monotonicity property of justifi... |

117 | Explicit provability and constructive semantics
- Artemov
(Show Context)
Citation Context ...roofs) remained an impediment to both the formalizing of BrouwerHeyting-Kolmorogorov semantics of proofs and providing a long-anticipated exact provability semantics for Gödel’s provability logic S4 (=-=[3; 4; 6; 61]-=-). This lack of a justification component has, perhaps, contributed to a certain gap between epistemic logic and mainstream epistemology ([31; 32]). We wish to think that Justification Logic is a step... |

117 |
The Logic of Provability
- Boolos
- 1993
(Show Context)
Citation Context ...hich are based on exhaustive search. It would be a mistake to draw a conclusion that any modal logic has a reasonable Justification Logic counterpart. For example, the logic of formal provability GL (=-=[8; 14]-=-) contains the Löb Principle ✷(✷F →F )→✷F, (43) which does not seem to have an epistemically acceptable explicit version. Let us consider, for example, a case when F is the propositional constant ⊥ fo... |

110 |
Sense and Reference
- FREGE
- 1960
(Show Context)
Citation Context ...us Ponens, qfJ proves c:{(f = g)→[P (f)→P (g)]}. c:{(f = g)→[P (f)→P (g)]}→{u:(f = g)→(c·u):[P (f)→P (g)]}. u:(f = g)→(c·u):[P (f)→P (g)]. It suffices now to pick c·u as s(u). ✷ However, Frege cases (=-=[25]-=-) can be represented in qfJ: an unjustified substitution can fail in qfJ. Namely, for any individual variables x and y, a predicate symbol P , and justification term s, the formula (x = y)→s:[P (x)↔P ... |

95 |
Introduction to Mathematical Philosophy
- Russell
- 1919
(Show Context)
Citation Context ...ket’ has been confirmed in our formal model of Case I. 6.2 Eliminating Definite Description, Russell’s style We can eliminate definite descriptions from Case I using, e.g., Russell’s translation (cf. =-=[24; 48; 54; 55]-=-) of definite descriptions. According to Russell, C(ιxJ(x)) contains a hidden uniqueness assumption and reads as ∃x[J(x) ∧ ∀y(J(y)→y = x) ∧ C(x)], (20) and Jones = ιxJ(x) as J(Jones) ∧ ∀y(J(y)→y = Jon... |

93 |
Impossible possible worlds vindicated
- Hintikka
- 1975
(Show Context)
Citation Context ... in formal epistemology. However, the principle (12) smuggles the logical omniscience defect into modal epistemic logic, which does not have the capacity to measure how hard it is to attain knowledge =-=[18; 19; 35; 47; 51]-=-. Justification Logic provides natural means of escaping logical omniscience by keeping track of the size of evidence terms [10]. 3.2 Monotonicity of Justification The Monotonicity property of justifi... |

90 | The semantics of reflected proof - Allen, Constable, et al. - 1990 |

86 |
Intuitionnistic logic
- Dalen
- 1986
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Citation Context ...roofs) remained an impediment to both the formalizing of BrouwerHeyting-Kolmorogorov semantics of proofs and providing a long-anticipated exact provability semantics for Gödel’s provability logic S4 (=-=[3; 4; 6; 61]-=-). This lack of a justification component has, perhaps, contributed to a certain gap between epistemic logic and mainstream epistemology ([31; 32]). We wish to think that Justification Logic is a step... |

85 |
On the interpretation of intuitionistic number theory
- Kleene
- 1945
(Show Context)
Citation Context ...oofs. It can be traced back to Brouwer-Heyting-Kolmogorov informal semantics for intuitionistic logic [60], typed combinatory logic (cf. [59]), Kleene realizability semantics for intuitionistic logic =-=[36]-=-, Gödel’s logic of provability [27], and Logic of Proofs [2; 4; 28]. There are several natural interpretations of Justification Logic. Justification assertions of the format t:F read generically as t ... |

80 |
Ein interpretation des intuitionistischen Aussagenkalküls, Ergebnisse eines mathematischen Kolloquiums 4
- Gödel
- 1933
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Citation Context ...wer-Heyting-Kolmogorov informal semantics for intuitionistic logic [60], typed combinatory logic (cf. [59]), Kleene realizability semantics for intuitionistic logic [36], Gödel’s logic of provability =-=[27]-=-, and Logic of Proofs [2; 4; 28]. There are several natural interpretations of Justification Logic. Justification assertions of the format t:F read generically as t is a justification of F. (2) There ... |

66 | Operational modal logic
- Artemov
- 1995
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Citation Context ...ormal semantics for intuitionistic logic [60], typed combinatory logic (cf. [59]), Kleene realizability semantics for intuitionistic logic [36], Gödel’s logic of provability [27], and Logic of Proofs =-=[2; 4; 28]-=-. There are several natural interpretations of Justification Logic. Justification assertions of the format t:F read generically as t is a justification of F. (2) There is also a more strict ‘justifica... |

58 | The logic of proofs, semantically
- Fitting
- 2005
(Show Context)
Citation Context ...F may need some Constant Specification CS ′ which is different from CS. ✷ 4 Basic Epistemic Semantics The standard epistemic semantics for J has been provided by a proper adaptation of Fitting models =-=[22]-=- and Mkrtychev models [46]. 10A Fitting J-model M = (W, R, A, ⊩) is an arbitrary Kripke model (W, R, ⊩) supplied with an admissible evidence function A such that A(t, F ) ⊆ W for any justification t ... |

52 | Thought and knowledge
- Halpern
- 1996
(Show Context)
Citation Context ... in formal epistemology. However, the principle (12) smuggles the logical omniscience defect into modal epistemic logic, which does not have the capacity to measure how hard it is to attain knowledge =-=[18; 19; 35; 47; 51]-=-. Justification Logic provides natural means of escaping logical omniscience by keeping track of the size of evidence terms [10]. 3.2 Monotonicity of Justification The Monotonicity property of justifi... |

46 | Justified common knowledge
- Artemov
(Show Context)
Citation Context ...D4.” 33appeared independently in [50] and [52; 53], and JD45 in [50]. J45 has, perhaps, first been considered in this paper. Systems combining epistemic modalities and justifications were studied in =-=[5; 11; 12]-=-. Mkrtychev semantics for J, JT, and J4 with Completeness Theorem were found in [41]. Complexity bounds for LP and J4 were found [41; 45]. 9 Forgetful Projection and the Correspondence Theorem An intu... |

38 |
Models for the logic of proofs
- Mkrtychev
- 1997
(Show Context)
Citation Context ...pecification CS ′ which is different from CS. ✷ 4 Basic Epistemic Semantics The standard epistemic semantics for J has been provided by a proper adaptation of Fitting models [22] and Mkrtychev models =-=[46]-=-. 10A Fitting J-model M = (W, R, A, ⊩) is an arbitrary Kripke model (W, R, ⊩) supplied with an admissible evidence function A such that A(t, F ) ⊆ W for any justification t and formula F . Informally... |

37 |
An essay on Modal Logic
- Wright
- 1955
(Show Context)
Citation Context ... until 1963 when a paper by Edmund Gettier [26] opened the door to a broad philosophical discussion of the subject (cf. [17; 29; 42; 49; 57] and many others). Meanwhile, commencing from seminal works =-=[34; 62]-=-, the notions of Knowledge and Belief have acquired formalization by means of modal logic with atoms KF (F is known) and BF (F is believed). Within this approach, the following analysis is adopted: F ... |

34 |
A Causal Theory of Knowing
- Goldman
- 1967
(Show Context)
Citation Context ...s Justified True Belief in the dialogues Theaetetus and Meno was widely accepted until 1963 when a paper by Edmund Gettier [26] opened the door to a broad philosophical discussion of the subject (cf. =-=[17; 29; 42; 49; 57]-=- and many others). Meanwhile, commencing from seminal works [34; 62], the notions of Knowledge and Belief have acquired formalization by means of modal logic with atoms KF (F is known) and BF (F is be... |

33 |
Introducing justification into epistemic logic
- Artemov, Nogina
(Show Context)
Citation Context ...tions are not always assumed to be factive. 4. In this paper, we consider the case of one agent only, although several examples of multi-agent Justification Logic systems have already been developed (=-=[5; 12; 56]-=-). Formal logical methods do not directly solve philosophical problems, but rather provide a tool for analyzing assumptions and making sure that we draw correct conclusions. Our hope is that Justifica... |

33 |
On the model theory of knowledge
- McCarthy, Sato, et al.
- 1978
(Show Context)
Citation Context ...ng the problem of logical omniscience; • evidence-based approach to Common Knowledge (so-called Justified Common Knowledge) which provides a rigorous semantics to McCarthy’s ‘any fool knows’ systems (=-=[1; 5; 43]-=-). Justified Common Knowledge offers formal systems which are less restrictive than the usual epistemic logics with Common Knowledge [5]. 4. It remains to be seen to what extent Justification Logic ca... |

26 |
Belief, awareness, and limited reasoning: preliminary report
- Fagin, Halpern
- 1985
(Show Context)
Citation Context |

22 |
On explicit counterparts of modal logics
- Brezhnev
- 2000
(Show Context)
Citation Context ...mic semantics and completeness [21; 22] were first established for LP. A fair amount of work has already been done on Jusification Logics other then LP. Systems J, J4, and JT were first considered in =-=[15]-=- under different names and in a slightly different setting 9 . JT45 9 [15] also considered variants of Justification Logic systems which in our notations would be called “JD” and “JD4.” 33appeared in... |

21 | On epistemic logic with justification
- Artemov, Nogina
- 2005
(Show Context)
Citation Context ...le played by LP for Justification Logic, we suggest keeping the name LP for this system. 8 A reasonable proof-compliant way to represent negative introspection in Justification Logic was suggested in =-=[9]-=-, but we will not consider it here. 31and JT45 = J45 + A4. Theorem 9 J4, LP, J45, JD45, and JT45 enjoy the Deduction Theorem, closure under substitution, and Internalization: if ⊢ F then ⊢ p:F , for ... |

21 |
Vortrag bei Zilsel
- Gödel
- 1938
(Show Context)
Citation Context ...ormal semantics for intuitionistic logic [60], typed combinatory logic (cf. [59]), Kleene realizability semantics for intuitionistic logic [36], Gödel’s logic of provability [27], and Logic of Proofs =-=[2; 4; 28]-=-. There are several natural interpretations of Justification Logic. Justification assertions of the format t:F read generically as t is a justification of F. (2) There is also a more strict ‘justifica... |

20 | On the first-order logic of proofs
- Artemov, Yavorskaya
(Show Context)
Citation Context ...that the information about Kripke structure in Fitting models can be completely encoded by the admissible evidence function. Mkrtychev models play an important theoretical role in Justification Logic =-=[7; 16; 38; 41; 45]-=-. On the other hand, as we will see in Section 6, Fitting models can be useful a counter-models with desired properties as they take into account both epistemic Kripke structure and evidence structure... |

19 |
On the complexity of explicit modal logics
- Kuznets
- 2000
(Show Context)
Citation Context ...that the information about Kripke structure in Fitting models can be completely encoded by the admissible evidence function. Mkrtychev models play an important theoretical role in Justification Logic =-=[7; 16; 38; 41; 45]-=-. On the other hand, as we will see in Section 6, Fitting models can be useful a counter-models with desired properties as they take into account both epistemic Kripke structure and evidence structure... |

18 | The basic logic of proofs
- Artëmov, Straßen
- 1993
(Show Context)
Citation Context ...ty. However, it is an intriguing challenge to develop a theory of non-monotonic justifications which prompt belief revision. Some Justification Logic systems without Monotonicity have been studied in =-=[13; 39; 40]-=-. 3.3 Basic Justification Logic J0 Justification terms (polynomials) are built from justification variables x, y, z, . . . and justification constants a, b, c, . . . by means of the operations applica... |

18 |
Knowledge: undefeated justified true belief
- Lehrer, Paxson
- 1968
(Show Context)
Citation Context ...s Justified True Belief in the dialogues Theaetetus and Meno was widely accepted until 1963 when a paper by Edmund Gettier [26] opened the door to a broad philosophical discussion of the subject (cf. =-=[17; 29; 42; 49; 57]-=- and many others). Meanwhile, commencing from seminal works [34; 62], the notions of Knowledge and Belief have acquired formalization by means of modal logic with atoms KF (F is known) and BF (F is be... |

16 |
Mainstream and Formal Epistemology
- Hendricks
- 2006
(Show Context)
Citation Context ...lity semantics for Gödel’s provability logic S4 ([3; 4; 6; 61]). This lack of a justification component has, perhaps, contributed to a certain gap between epistemic logic and mainstream epistemology (=-=[31; 32]-=-). We wish to think that Justification Logic is a step towards filling this void. The contribution of this paper to epistemology can be briefly summarized as follows. We argue that justifications have... |

16 | A note on some explicit modal logics
- Pacuit
- 2005
(Show Context)
Citation Context ...nstead of a:(b:(c:A)), etc. Such modifications are minor and they do not affect the main theorems and applications of Justification Logic. 8.2 Negative Introspection Pacuit and Rubtsova considered in =-=[50; 52; 53]-=- the Negative Introspection operation ‘?’ which verifies that a given justification assertion is false. A possible motivation for considering such an operation could be that the positive introspection... |

15 | On the complexity of the reflected logic of proofs
- Krupski
(Show Context)
Citation Context ...that the information about Kripke structure in Fitting models can be completely encoded by the admissible evidence function. Mkrtychev models play an important theoretical role in Justification Logic =-=[7; 16; 38; 41; 45]-=-. On the other hand, as we will see in Section 6, Fitting models can be useful a counter-models with desired properties as they take into account both epistemic Kripke structure and evidence structure... |

14 | Making knowledge explicit: how hard it is
- Brezhnev, Kuznets
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14 |
A semantics for the logic of proofs
- Fitting
- 2003
(Show Context)
Citation Context ...properties of Justification Logic as internalization, realization, arithmetical semantics [2; 4], symbolic models and complexity estimates ([16; 41; 45; 46]), and epistemic semantics and completeness =-=[21; 22]-=- were first established for LP. A fair amount of work has already been done on Jusification Logics other then LP. Systems J, J4, and JT were first considered in [15] under different names and in a sli... |

13 |
belief and counterfactual reasoning in games
- Knowledge
- 1996
(Show Context)
Citation Context |

12 |
Justified and common knowledge: Limited conservativity
- Antonakos
- 2007
(Show Context)
Citation Context ...ng the problem of logical omniscience; • evidence-based approach to Common Knowledge (so-called Justified Common Knowledge) which provides a rigorous semantics to McCarthy’s ‘any fool knows’ systems (=-=[1; 5; 43]-=-). Justified Common Knowledge offers formal systems which are less restrictive than the usual epistemic logics with Common Knowledge [5]. 4. It remains to be seen to what extent Justification Logic ca... |

12 |
Logic of knowledge with justifications from the provability perspective
- Artemov, Nogina
- 2004
(Show Context)
Citation Context ...D4.” 33appeared independently in [50] and [52; 53], and JD45 in [50]. J45 has, perhaps, first been considered in this paper. Systems combining epistemic modalities and justifications were studied in =-=[5; 11; 12]-=-. Mkrtychev semantics for J, JT, and J4 with Completeness Theorem were found in [41]. Complexity bounds for LP and J4 were found [41; 45]. 9 Forgetful Projection and the Correspondence Theorem An intu... |

12 |
Evidence reconstruction of epistemic modal logic S5
- Rubtsova
- 2006
(Show Context)
Citation Context ...nstead of a:(b:(c:A)), etc. Such modifications are minor and they do not affect the main theorems and applications of Justification Logic. 8.2 Negative Introspection Pacuit and Rubtsova considered in =-=[50; 52; 53]-=- the Negative Introspection operation ‘?’ which verifies that a given justification assertion is false. A possible motivation for considering such an operation could be that the positive introspection... |

11 |
The single-conclusion proof logic and inference rules specification
- Krupski
- 2001
(Show Context)
Citation Context ...ty. However, it is an intriguing challenge to develop a theory of non-monotonic justifications which prompt belief revision. Some Justification Logic systems without Monotonicity have been studied in =-=[13; 39; 40]-=-. 3.3 Basic Justification Logic J0 Justification terms (polynomials) are built from justification variables x, y, z, . . . and justification constants a, b, c, . . . by means of the operations applica... |

10 | Logical omniscience via proof complexity
- Artemov, Kuznets
(Show Context)
Citation Context ... capacity to measure how hard it is to attain knowledge [18; 19; 35; 47; 51]. Justification Logic provides natural means of escaping logical omniscience by keeping track of the size of evidence terms =-=[10]-=-. 3.2 Monotonicity of Justification The Monotonicity property of justifications has been expressed by the operation sum ‘+’ which can be read from (10). If s:F , then whichever evidence t occurs, the ... |

9 |
F.: 2003, Active agents
- Hendricks
(Show Context)
Citation Context ...lity semantics for Gödel’s provability logic S4 ([3; 4; 6; 61]). This lack of a justification component has, perhaps, contributed to a certain gap between epistemic logic and mainstream epistemology (=-=[31; 32]-=-). We wish to think that Justification Logic is a step towards filling this void. The contribution of this paper to epistemology can be briefly summarized as follows. We argue that justifications have... |

7 | Referential logic of proofs
- Krupski
(Show Context)
Citation Context ...ty. However, it is an intriguing challenge to develop a theory of non-monotonic justifications which prompt belief revision. Some Justification Logic systems without Monotonicity have been studied in =-=[13; 39; 40]-=-. 3.3 Basic Justification Logic J0 Justification terms (polynomials) are built from justification variables x, y, z, . . . and justification constants a, b, c, . . . by means of the operations applica... |

7 |
Derivability in certain subsystems of the Logic of Proofs is Π p 2-complete
- Milnikel
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5 |
On realization of S5-modality by evidence terms
- Rubtsova
(Show Context)
Citation Context ...nstead of a:(b:(c:A)), etc. Such modifications are minor and they do not affect the main theorems and applications of Justification Logic. 8.2 Negative Introspection Pacuit and Rubtsova considered in =-=[50; 52; 53]-=- the Negative Introspection operation ‘?’ which verifies that a given justification assertion is false. A possible motivation for considering such an operation could be that the positive introspection... |

5 |
Multi-agent explicit knowledge
- Yavorskaya
- 2006
(Show Context)
Citation Context ...tions are not always assumed to be factive. 4. In this paper, we consider the case of one agent only, although several examples of multi-agent Justification Logic systems have already been developed (=-=[5; 12; 56]-=-). Formal logical methods do not directly solve philosophical problems, but rather provide a tool for analyzing assumptions and making sure that we draw correct conclusions. Our hope is that Justifica... |

4 | Reflective λ-calculus - Alt, Artemov - 2001 |